There are 351 combinations of numbers. Some are obviously worth less than others. For example, I looked at the end to see if a string had 6 or less numbers above 40, 6 or more single digit numbers. If both of those things are true, the sum is probably below 700.
Using only that eye test, it becomes clear that most of the strings of numbers will not have a sum below 700. Since we're only looking for the three lowest sums, they're not that hard to find.